Example on Root Locus Sketching and Control Design


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1 Example on Root Locus Sketching and Control Design MCE44  Spring 5 Dr. Richter April 25, 25 The following figure represents the system used for controlling the robotic manipulator of a Mars Rover. We wish to: R(s) + K(s s+2) s U(s) (s+)(s+2) Y (s) Figure : Control System. Provide a detailed handsketch of the root locus. 2. Find the gain K so that the system responds with an overshoot of percent. 3. Find the gain K so that the system responds with an overshoot of less than percent and a settling time as fast as possible. Root Locus Sketching The poles of the openloop transfer function G(s)K(s) are given by and there is a pair of complex zeros at s =, s =, s = 2 s = 3.25 ±.99i The order of the system (number of poles) is n = 3 and the number of zeros is m = 2. According to this, m = 2 poles go to the zeros as K and the remaining pole goes to infinity following an asymptote. The angle of the asymptote is obviously 8, but in general, we find asymptote angles with the formula φ A = 8 (2k + ) n m where k =,,..n m. In our case, n m =, so φ A = 8 Again, in this case it is unnecessary to find the center of the asymptotes, but in the general case it is obtained by subtracting the sum of zeros from the sum of poles and dividing by n m. In this case σ A = ( 2) ( i i) With this information, we still have two possibilities for the shape of the root locus that comply with the basic requirements, as shown in the figure below. In order to determine which one of the possibilities is = 3.5
2 Possibility Possibility 2 Figure 2: Two possibilities for the root locus correct, we find the breakin and breakaway points. To do this, solve for K from the characteristic equation and call the result p(s): + K s s )(s + 2) K = s(s s(s + )(s + 2) s s + 2 = p(s) Now differentiate p(s) and equate it to zero to find maxima: ( dp(s) (s s + 2)(3s 2 + 6s + 2) (2s + 6.5)(s 3 + 3s 2 ) + 2s) = ds (s s 2 + 2) 2 = which reduces to the fourthorder polynomial equation: with roots s 4 + 3s s s + 24 = s = 5.78, s = 5.47, s =.658, s =.4933 Since the point s =.658 is not part of the locus, we discard it. The other three points are valid, so we can determine that the sketch on the right of Fig. 2 is the correct one. Although it is not necessary here, we will show how to determine the angles of arrival to the complex zeros. The angle of arrival at a given zero is calculated with θ a = 8 + angles from poles angles from the other zeros The following schematic illustrates the calculation: The angle of arrival is calculated as θ a = = This result must be reduced by an integer number of turns, however. The angle can be expressed as = so that is the meaningful arrival angle. The rootlocus can be obtained in one step by using Matlab: >> num=[ 6.5 2]; >> den=[ 3 2 ]; >> rlocus(num,den) Now we determine the gain K so that the step response has an overshoot of percent. To do this, we notice that the corresponding damping ratio is ζ =.82. The angle of the dominant poles must then be φ = cos (.82) =.694 radians, which is We draw a line forming that angle with the negative 2
3 Angle of arrival to complex zero= = Remove integer multiples of 36 to get Figure 3: Calculating the angles of arrival and departure real axis and look for intersections with the root locus. There are two intersections. We can find the gain and poles of the intersections interactively with Matlab. The intersections are s = 4.7 ± 3.29i, at K =. s =.492 ±.344i, at K =.624 However, for K =. no pole dominance occurs. Indeed, the other root lies at s = 3.67, very close to the complex pair. In addition, the zeros are expected to have an influence, as they have a relatively large time constant. For K =.624, the other pole is very close to its openloop location of 2. This pole and the complex pair of zeros have time constants of /2=.5 and /3.25=.37, while the complex pair of poles have a time constant of /.492= The complex poles are 4 times slower than the zeros and 6.5 times slower than the real pole. Dominance is not achieved according to the factor of 8 criterion, but we can expect some approximation to the percent overshoot. Figure 5 shows the responses with approximately percent overshoot and a slow response when K =.624, and 5 percent overshoot for K =.. Now we wish to select a gain that has a low overshoot but that produces a fast response. By examining the root locus, we see that as the gain increases, the closedloop complex poles approach the zeros, while the other root stays real and moves to the left. At K =, the complex poles match the zeros and a stable polezero cancellation occurs. When we design control systems, polezero cancellations happening on the lefthalf complex plane are allowed, while those happening on the imaginary axis or on the righthalf of the complex plane are not. The reason for this is as follows: If in order to cancel a pair of righthalf plane zeros of the plant we introduce a pair of matching righthalf plane poles, we have a stable closedloop system but an unstable controller. Physically, very large electrical signals or mechanical displacements will occur in the elements realizing the control, leading to system failure. In our case the polezero cancellation at high gain happens on the lefthalf plane, so using a high gain is not ruled out. The actual value of the gain to attain the desired overshoot is best found by trialanderror simulation in Matlab. The value K = 3 places the poles at s = , s = 3.27 ±.2i 3
4 Root Locus 3 2 Imag Axis Real Axis Figure 4: Root Locus and Constant Damping Line The closedloop transfer function is Y (s) R(s) = 3 (s i)(s i) (s i)(s i)(s ) Note that the complex poles and the zeros almost cancel each other. Figure 6 shows the resulting step response. As a final observation, the saying there s no free lunch applies here. Such a high performance is obtained at the cost of very high control effort. A direct simulation can show that output of the controller for K = 3 is four orders of magnitude larger than that for K =.624. A realistic controller design must take into account limitations in the control magnitude. 4
5 .5 Responses for K=. and K=.642 K= K= Figure 5: Responses for percent overshoot 5
6 .4 Response with K= Figure 6: Final Response 6
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